## physicsdiff.mpl Version 0.2

This procedure allows you to perform a first variation of some expression (presumed to be inside an integral, e.g. the Lagrangian density in an action). It will differentiate the expression treating the specified function as the differentiation parameter, with the additional feature that it automatically attempts to integrate first order derivatives of that function by parts.
To call the procedure, run:
```
physicsdiff(expression,function,diffparam);
```

### Example

```
expr := h(r)*g(r)*2 + h(r)*diff(h(r),r)^2*3 + diff(h(r),r):
physicsdiff(expr,h(r),r);
2*g(r) + 3*(diff(h(r),r))^2 - 2*diff(3*h(r)*diff(h(r),r),r)
```

### Source Code

```
#This maple procedure does a functional variation, much like diff does when used
#with(Physics):, but with the added feature that it can vary simple derivatives
#with respect to that function, ie. vary(c*diff(f(r),r)^2,f(r)) using
#integration by parts.
_physicsdiff2_term := proc(function,term,innerterm,dparam)
local counter, thecoeff,tmp;
if is(op(0,innerterm) = `^`) then
#In this case, the term is a power of a derivative of the function of interest
thecoeff := coeff(term,diff(function,dparam)^(op(2,innerterm)));
#print(term,thecoeff,innerterm,"isntdiff");
RETURN(op(2,innerterm)*diff(thecoeff*diff(function,dparam)^(op(2,innerterm)-1),dparam));
elif is(op(0,innerterm)=`diff`) then
#Here the function of interest is alone
thecoeff := coeff(term,diff(function,dparam));
#print(term,innerterm,"isdiff");
#RETURN(diff(thecoeff,dparam)*function);
#We can't return the function with it! That is the part that gets differentiated away!
RETURN(diff(thecoeff,dparam));
else
print("Error - could not determine power of term.");
RETURN(0);
fi:
end proc:
_physicsdiff_term := proc(expr,function,term,dparam)
local counter,counter2,tmp,retterm;
retterm := 0;
if is(op(0,dparam)=`list`) then
for counter from 1 to nops(dparam) do
if (has(term,diff(function,dparam[counter])) and not(has(term,diff(function,dparam[counter] $ 2)))) then
if is(op(0,term)=`*`) then
for counter2 from 1 to nops(term) do
tmp := op(counter2,term);
if (has(tmp,diff(function,dparam[counter]))) then
retterm := retterm - _physicsdiff2_term(function,term,tmp,dparam[counter]);
fi:
end do:
elif is(op(0,term)=`^`) then
tmp := term;
if(has(tmp,diff(function,dparam[counter]))) then
retterm := retterm - _physicsdiff2_term(function,term,tmp,dparam[counter]);
fi:
else
print("Found a term that's just a lone derivative. Integration by parts failed.");
fi:
RETURN(retterm);
elif has(term,diff(function,dparam[counter] $ 2)) then
print("Error, found a derivative of order greater than 2. This term is killed.");
RETURN(0);
fi:
end do:
RETURN(0);
else
if (has(term,diff(function,dparam)) and not(has(term,diff(function,dparam $ 2)))) then
if is(op(0,term)=`*`) then
for counter from 1 to nops(term) do
tmp := op(counter,term);
if (has(tmp,diff(function,dparam))) then
retterm := retterm - _physicsdiff2_term(function,term,tmp,dparam);
fi:
end do:
elif is(op(0,term)=`^`) then
print("Found a term that has powers.");
tmp := term;
if(has(tmp,diff(function,dparam))) then
retterm := retterm - _physicsdiff2_term(function,term,tmp,dparam);
fi:
else
#There is nothing to be done - the term is just a lone derivative of "function" - it will be killed.
print("Found a term that's just a lone derivative. Integration by parts failed.");
fi:
RETURN(retterm);
elif has(term,diff(function,dparam $ 2)) then
print("Error, found a derivative of order greater than 2. This term is killed.");
RETURN(0);
else
RETURN(0);
fi:
fi:
end proc:
physicsdiff := proc(expr,function,dparam)
local expr2,newexpr,exprlength,term,tmp,counter;
description "This Maple procedure will perform a physicists' functional differentiation, neglecting boundary terms and integrating by parts to functionally vary an expression with respect to a function, even if it contains powers of first derivatives of that function. Written by W. Brenna, Sept 3 2012. Version 0.2.";
if not(is(op(0,f(r)*g(r))=`*`)) then
print("Error: Conflicting package detected! The `*` declaration has been redefined and results are therefore unreliable. Make sure you have loaded physicsdiff before running with(Physics).");
RETURN(expr);
fi:
expr2 := expand(expr);
exprlength := nops(expr2);
#newexpr := expr2;
newexpr := 0;
if is(op(0,expr2)=`+`) then
for counter from 1 to exprlength do
term := op(counter,expr2);
#This just does the first order integration by parts.
newexpr := newexpr + _physicsdiff_term(newexpr,function,term,dparam):
end do:
else
term := expr2;
newexpr := newexpr + _physicsdiff_term(newexpr,function,term,dparam):
fi:
#Finally, we do the regular differentiation by parts to build up the final
#terms.
#RETURN(newexpr + frontend(diff,[expr2,function]));
#frontend is hopeless for this - non-integer powers make it cry.
#print(Physics:-diff(expr2,function));
RETURN(newexpr + Physics:-diff(expr2,function));
end proc:
```